iPad Mini 4 Charger/Lighting port possible damage

Did you try to also purchase a new cable? Sometimes the cables get a kink and after time the wires inside have intermittent connection issues.

Did you purchase an original Apple iPad charger?

http://www.apple.com/ca/shop/product/MD836LL/A/apple-12w-usb-power-adapter?fnode =91

Looks like this:

Apple specifically says not to use compressed air:

iPad

Handle your iPad with care to maintain its appearance. If you're concerned about scratching or abrasion, you can use one of the many cases sold separately. To clean iPad, unplug all cables and turn off iPad (press and hold the Sleep/Wake button, and then slide the onscreen slider). Use a soft, slightly damp, lint-free cloth. Avoid getting moisture in openings. Don't use window cleaners, household cleaners, compressed air, aerosol sprays, solvents, ammonia, abrasives, or cleaners containing hydrogen peroxide to clean iPad. iPad has an oleophobic coating on the screen; simply wipe iPad's screen with a soft, lint-free cloth to remove oil left by your hands. The ability of this coating to repel oil will diminish over time with normal usage, and rubbing the screen with an abrasive material will further diminish its effect and might scratch your screen.

Cleaning your Apple products - Apple Support

(bold mine)

You can still take it to the Apple Store Genius Bar and let them look at it. If the hints people have suggested here still don't work, there probably is some sort of malfunction of the port. That being said, if you bring it in to have them look at it, they aren't going to charge you for that. When you would get charged is if they have to do any work or replacement of it. They will be able to tell you at the time what kind of charges you are looking at, but the initial inspection of it you won't be charged for.

Examine the charging port under a bright light, preferably with a magnifier. Remove any lint or other debris using a non-conductive tool such as a wooden toothpick or the stick from a cotton swab.

Try cleaning the charging port with compressed air to dislodge dust or lint.

Did you take it to apple to be looked at? How much is it? I'm having the same problem but I don't have insurance either.